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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Trees and branches in Banach spaces

Authors: E. Odell and Th. Schlumprecht
Journal: Trans. Amer. Math. Soc. 354 (2002), 4085-4108
MSC (2000): Primary 46B03
Published electronically: May 20, 2002
MathSciNet review: 1926866
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Abstract: An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree $\mathcal{T}$of a certain type on a space $X$ is presumed to have a branch with some property. It is shown that then $X$ can be embedded into a space with an FDD $(E_i)$ so that all normalized sequences in $X$ which are almost a skipped blocking of $(E_i)$ have that property. As an application of our work we prove that if $X$ is a separable reflexive Banach space and for some $1<p<\infty$ and $C<\infty$ every weakly null tree $\mathcal{T}$ on the sphere of $X$ has a branch $C$-equivalent to the unit vector basis of $\ell_p$, then for all $\varepsilon>0$, there exists a subspace of $X$ having finite codimension which $C^2+\varepsilon$ embeds into the $\ell_p$ sum of finite dimensional spaces.

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Additional Information

E. Odell
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712

Th. Schlumprecht
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Received by editor(s): October 10, 2000
Received by editor(s) in revised form: November 7, 2001
Published electronically: May 20, 2002
Additional Notes: Research supported by NSF
Article copyright: © Copyright 2002 American Mathematical Society